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Let (an)nen be a bounded sequence in R. For all n e N define bn =...
18. If ai, az, as,... is a bounded sequence of real numbers, define lim sup an...
18. If ai, az, as,... is a bounded sequence of real numbers, define lim sup an (also denoted lim an) to be --+ n+ l.u.b. {z ER: an > & for an infinite number of integers n} and define lim inf an (also denoted lim an) to be g.l.b. {ER: An <for an infinite number of integers n}. Prove that lim inf an Slim sup an, with the equality holding if and only if the sequence converges. 19. Let ai,...
Q2 (10 points) Vn2 + 4 – n, n E N. 2. Let (an) neN be...
Q2 (10 points) Vn2 + 4 – n, n E N. 2. Let (an) neN be the sequence with a, (a) Prove that lim,→0 an 0. lim,-00 bn, and prove the limit exists, by using the definition. (b) Let bn = n an . Find L =
13 14 Exercise 13: Let (xn) be a bounded sequence a S be the set of...
13 14
Exercise 13: Let (xn) be a bounded sequence a S be the set of limit points of (n), i.e. S:{xER there exists a subsequence () s.t. lim } ko0 Show lim inf inf S n-o0 Hint: See lecture for proof lim sup Exercise 14: (Caesaro revisited) Let (x) be a convergent sequence. Let (yn) be the sequence given by Yn= n for all n E N. Show that lim sup y lim sup n n-+00 n o0
Let (xn) be a bounded sequence of real numbers, and put u = lim supn→∞ xn . Let E be the set consisting of the limits of all convergent subsequences of (xn). Show that u ∈ E and that u = sup(E). Form...
Let (xn) be a bounded sequence
of real numbers, and put u = lim supn→∞ xn . Let E be the set
consisting of the limits of all convergent subsequences of (xn).
Show that u ∈ E and that u = sup(E).
Formulate and prove a similar result for lim infn→∞ xn .
Thank you!
7. Let (Fm) be a bounded sequence of real numbers, and put u-lim supn→oorn . Let E be the set consisting of the limits of...
Exercise 2.3.9. (a) Let (an) be a bounded (not necessarily convergent) sequence, and assume lim bn...
Exercise 2.3.9. (a) Let (an) be a bounded (not necessarily convergent) sequence, and assume lim bn = 0. Show that lim(anon) = 0. Why are we not allowed to use the Algebraic Limit Theorem to prove this?
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r...
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r is bounded above, x = lim sup (r) if and only if For all 0 there is an NEN, such that x <x+e whenevern > N, and b. For all >0 and all M, there is n > M with x - e< In a.
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r is bounded above,...
Separate each answer? 5) Define the supremum of a bounded above set SCR. 6) Define the...
Separate each answer?
5) Define the supremum of a bounded above set SCR. 6) Define the infimum of a bounded below set SCR. 7) Give the completeness property of R 8) Give the Archimedean property of R. 9) Define a density set of R. 10) Define the convergence of a sequence of R and its limit. 11) State the Squeeze theorem for the convergent sequence. 12) Give the definition of increasing sequence, decreasing sequence, monotone se- quence. 13) Give the...
(a) Suppose lim(Fm) = 1. Prove or disprove: There exists no E N such that IFml > 0.99 for all o (...
The work provided for part (b) was not correct.
(a) Suppose lim(Fm) = 1. Prove or disprove: There exists no E N such that IFml > 0.99 for all o (b) Prove or disprove:If (an) converges to a non-zero real number and (anbn) is convergent, then (bn) is convergent. RUP ) Let an→ L,CO) and an bn→12 n claim br) comvetgon Algebra of sesuenes an
(a) Suppose lim(Fm) = 1. Prove or disprove: There exists no E N such that...
4. A function sequence (fn)neN İs unifonnly bounded on D if there is an M0 such that If, (xl s M ...
4. A function sequence (fn)neN İs unifonnly bounded on D if there is an M0 such that If, (xl s M for allxin D and all in N. Show that if s uniformly convergent on D and each fn is bounded on D, then (fn)neN 1S uniformly bounded on D. Use this to conclude that the function sequence in Example 8.3 is not uniformly convergent. Example 8.3 Let f : (0.1)-R be given by falx)-nx(I-2for eaclh n in (Higure 8.2)...
1,2 Let (an)nen be a sequence of real numbers that is bounded from above. Consider L...
1,2
Let (an)nen be a sequence of real numbers that is bounded from above. Consider L := lim suPn7o An, prove that: For all e > 0 there are only finitely many n for which an > L + €. For all e > 0 there are infinitely many n for which an > L - €.
18. If ai, az, as,... is a bounded sequence of real numbers, define lim sup an (also denoted lim an) to be --+ n+ l.u.b. {z ER: an > & for an infinite number of integers n} and define lim inf an (also denoted lim an) to be g.l.b. {ER: An <for an infinite number of integers n}. Prove that lim inf an Slim sup an, with the equality holding if and only if the sequence converges. 19. Let ai,...
Q2 (10 points) Vn2 + 4 – n, n E N. 2. Let (an) neN be the sequence with a, (a) Prove that lim,→0 an 0. lim,-00 bn, and prove the limit exists, by using the definition. (b) Let bn = n an . Find L =
13 14
Exercise 13: Let (xn) be a bounded sequence a S be the set of limit points of (n), i.e. S:{xER there exists a subsequence () s.t. lim } ko0 Show lim inf inf S n-o0 Hint: See lecture for proof lim sup Exercise 14: (Caesaro revisited) Let (x) be a convergent sequence. Let (yn) be the sequence given by Yn= n for all n E N. Show that lim sup y lim sup n n-+00 n o0
Let (xn) be a bounded sequence
of real numbers, and put u = lim supn→∞ xn . Let E be the set
consisting of the limits of all convergent subsequences of (xn).
Show that u ∈ E and that u = sup(E).
Formulate and prove a similar result for lim infn→∞ xn .
Thank you!
7. Let (Fm) be a bounded sequence of real numbers, and put u-lim supn→oorn . Let E be the set consisting of the limits of...
Exercise 2.3.9. (a) Let (an) be a bounded (not necessarily convergent) sequence, and assume lim bn = 0. Show that lim(anon) = 0. Why are we not allowed to use the Algebraic Limit Theorem to prove this?
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r is bounded above, x = lim sup (r) if and only if For all 0 there is an NEN, such that x <x+e whenevern > N, and b. For all >0 and all M, there is n > M with x - e< In a.
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r is bounded above,...
Separate each answer?
5) Define the supremum of a bounded above set SCR. 6) Define the infimum of a bounded below set SCR. 7) Give the completeness property of R 8) Give the Archimedean property of R. 9) Define a density set of R. 10) Define the convergence of a sequence of R and its limit. 11) State the Squeeze theorem for the convergent sequence. 12) Give the definition of increasing sequence, decreasing sequence, monotone se- quence. 13) Give the...
The work provided for part (b) was not correct.
(a) Suppose lim(Fm) = 1. Prove or disprove: There exists no E N such that IFml > 0.99 for all o (b) Prove or disprove:If (an) converges to a non-zero real number and (anbn) is convergent, then (bn) is convergent. RUP ) Let an→ L,CO) and an bn→12 n claim br) comvetgon Algebra of sesuenes an
(a) Suppose lim(Fm) = 1. Prove or disprove: There exists no E N such that...
4. A function sequence (fn)neN İs unifonnly bounded on D if there is an M0 such that If, (xl s M for allxin D and all in N. Show that if s uniformly convergent on D and each fn is bounded on D, then (fn)neN 1S uniformly bounded on D. Use this to conclude that the function sequence in Example 8.3 is not uniformly convergent. Example 8.3 Let f : (0.1)-R be given by falx)-nx(I-2for eaclh n in (Higure 8.2)...
1,2
Let (an)nen be a sequence of real numbers that is bounded from above. Consider L := lim suPn7o An, prove that: For all e > 0 there are only finitely many n for which an > L + €. For all e > 0 there are infinitely many n for which an > L - €.
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